sure, you can use a "flat" or "uniform" prior. in cases where you have no other information you might have no choice but to use a flat prior. or heck you might insist that your model is more predictive when you use a flat prior instead of an x, y, or z prior. and some other person might disagree with you and say that his model using a, b, or c prior is better and more predictive than yours. and then you guys can solve the dispute by betting on it. in time you'll come to know whose model is better. or maybe you won't. maybe one of you will be old and dying and convinced that your model is actually better but that you've just run really bad for the past 60 years. and maybe you'd be right.

Ok, this was mostly what I was aiming at. Flat prior in case of winrate distributions cannot obviously be correct, but if we think about player, who studies poker intensively, then also a prior using the whole player pool, which to my belief consists mostly of players who don't study poker very intensively, might be equally misleading. OP calculated his "true winrate" using a flat prior (didn't he?), and I'd say it is an useful prediction, given big enough sample size. It might be even the best available, unless we have some insight about winrate distribution in some more accurate peer group.

Thank you once again for explaining this. Hopefully I now have some understanding on the subject.

Am I not using the confidence interval to say: We can be 95% confident that the population mean falls within the interval X +- Y? I set up the problem with the population mean being BB/100.

But everybody "feels" like they're awesome at Poker. It's just like sex: everyone thinks they're the best. If I couldn't figure out if I was a 10Bb/100 winner or a 0.1 BB/100 winner, but I knew that I was one of the two, then I'd still know that I was a winner.

But what I'm interested in is more KNOWING that I'm a winner based on the math, rather than hoping/wishing/thinking I'm a winner because I feel like I am better than everyone else (everybody thinks this except the completely casual player). Watch a little WSOP on TV and become an instant genius overnight.

It is also more fascinating to me because most people think that you can't know anything with a sample size of 1250 hands, but knowing whether or not you are a winning player or not is indeed something, in my opinion. Sure it's not something awesome like my true win rate. But it's always nice to know that you beat the game you play at over time. It's obvious to know if you are currently beating it, but I think that if you can be 90%-95% sure you should continue beating a game after 1250 hands instead of waiting for 10,000 hands like most people (aside from the people in this thread) think, then you're doing something good.

Only the completely clueless macho idiots "feel like they're awesome no matter what".

Its not that hard if you are honest with yourself. I have sat in games where I felt like I was the big fish (especially before I read anything about poker/ 2p2). I played against people who would effortlessly steal a pot from me if they felt i didn't connect, etc.

I'm not sure that I agree with this interpretation.

A 95% confidence interval means the following:

If we take a large number of samples of the same size from the same population and calculated a 95% confidence from each sample, then 95% of the intervals would include the true win rate, assumed to be a constant.

Now from a strictly theoretical point of view, the CI only tells you how ‘good’ the interval is. That is why the term confidence is used instead of probability. But, the older I get the less I worry about this theoretical issue. I think it is perfectly okay to interpret the interval as many do – namely as a "probabilistic" measure of the likely range of your true win rate.

Bayes allows you to let the true win rate have a distribution, unlike the frequentist approach. But, who determines if Bayes is applicable? You, the analyst, that’s who. So, if I can decide to use either the classical method with a fixed win rate or Bayes with a win rate distribution, why can’t I compromise and use the classical method with a quasi-varying win rate?

Let me summarize by asking the following:

Assume that you have a sample of 1,000,000 hands, instead of 1250, and you calculated a 99.99% confidence interval for win rate/100 and this interval turned out to be (3,5). Would you not conclude with very high certainty (confidence, if you prefer) that you were a a winning player? If your answer is yes, then I would say that OP's use of CI for 1250 hands and 95% confidence is the same idea but with less certainty in the conclusion.

agree completely. that is THE definition of a confidence interval.


could not disagree more.


well you can do whatever you want but it doesn't mean your model will be any good.



no, of course i wouldn't assume anything based on what a confidence interval told me - i think i've made that clear. now, after putting together my model it's possible this time i might end up with an answer that happens to fall in your range - i certainly acknowledge that possibility. sometimes the two methods will get you close to the same answer. but sometimes they won't. the devil's in the details. i'm looking for the right answer, and the methodology that best gets me there, period. (particularly if a lot of money is riding on it.)

statmanhal, let's say we do the following experiment:

we take a fair deck of 52 cards, shuffle it, and ask the subject to guess the top card. we write down his answer and then shuffle the card back into the deck and repeat the experiment 51 more times with the same subject, recording his answer everytime.

now let's say i come to you tomorrow and show you a guy who guessed 10 cards correctly (a 19.2% clip). note that his 99 percent confidence interval is [5.2%, 33.3%]. are you super confident that he has ability to guess the cards better than an average person? you gonna be willing to bet big money on his success rate in future trials? (e.g. if i set the line at 3.2% for his next batch of 52 cards you would confidently wager the over)

Your example is really not pertinent. In this case, we know what the actual guess probability is – 1/52, so assuming fair trials, your assumed results is something really unlikely to happen, and even if it did, of course I would not take the bet. See, I’m not against using prior distributions, but only when I can come up with a reasonable one, which in this case I can.

My point is that we are trying to get an estimate of a person’s win rate knowing little about the parameter. A confidence interval is a standard, well accepted measure for making inferences. We make judgments of much importance, such as clinical trials of potentially life saving drugs, using significance tests, a form of a confidence interval test in many cases. To reject the conclusion related to a confidence interval result after overwhelming evidence indicates the player has a winning record seems to me to be a bit too dogmatic in one’s Bayesian beliefs. But, let me quote a well-known author about that

“well you can do whatever you want but it doesn't mean your model will be any good.”

firstly, this is an awesome thread. well done guys.

i think a lot of the confusion that arises in this thread is because OP's numbers, on the surface, look reasonable. but the thinking is certainly flawed.

to illustrate this, i decided to tweak his session numbers a little bit.

for the purposes of this argument, lets assume that session nine didn't occur - OP felt a bit sick and decided to stay home instead of going to the casino.

lets also assume that on the very last hand of the last session, OP managed to pick up aces into kings and double. so instead of winning 81 BB, he wins 162BB.

both of those tweaks are within reason i think.

now, calculating the confidence interval, it has blown out to [-12.5, +46.26]

suddenly that looks horrible.

it would be very easy to make some very small tweaks to make it look even worse.

change the final session from a winning session to a losing one and the confidence interval as calculated above becomes about [-10 , +30]


small changes have a big effect, which is obvious but it also highlights that the thinking is flawed and it was just lucky that the original calculations turned out to be something approaching what you would intuitively think correct.

So you're saying..but what IF I won more money in less hands? You're saying that instead of having played 1250 hands, you can tell me that if my statistics only included 1150 hands with more money won then my calculations would be less reliable. But I have 1250 hands.. If I had 1150 hands then of course I'd be less confident in how I run (and I'm not saying that everyone can know in exactly 1250 hands how big of a loser they are). I think that certain play styles mean different things. If I was a hyper aggressive player I would have bigger swings than the ones that I have had. If I'd bought in with full $200 stacks I'd have seen bigger swings as well. But I have not bought in with big stacks. I've also (from these statistics) had -2 BI nights when I got it in ahead. Did I still include them in my statistics? Of course I did because I realize sometimes you can run bad too. Sometimes you win 2 BI's in a session, and sometimes you lose that many. That's probably why it says my win rate can be 19+- BB/100 than what it currently "is".

This is why I calculated that if I LOSE 2 buy-ins a night over the course of the next 5 nights in a row that I play, then I would have an average win-rate of -3 BB/100 hands over all of my hands played. This shows why the CI thinks that the worst that I'm doing is losing 3 BB/100 on average.

Also, I completely agree that this thread has been very awesome! I appreciate everyone's help so far. Can some of you Bayesian experts use my data to plug into a Bayesian formula and see what those results say?

#.......Profit......... Net Profit..Hrs....Tot. Hands....Session BB/100
12......$148......... $844.........50.......1250.............24.67
11......$(21)........ $696.........44.......1100............(5.25)
10..... $153..........$717.........40.......1000.......... ..38.25
9........$(100).......$564.........36.........900. ..........(33.33)
8........$49...........$664.........33.........825 .............10.89
7........$59...........$615.........28.5......713. ............14.75
6........$53...........$556.........24.5......613. .............13.25
5........$132..........$503........20.5.......513. ............33.00
4........$19............$371.......16.5........413 .............3.80
3........$226..........$352.......11.5........288. ............64.57
2........$(200)........$126.........8..........200 ............(50.00)
1........$326..........$326.........4..........100 .............81.50

If the winrate is a random variable, the average (or sum) win rate of each identical and independent session should be distributed normal w/ the mean centered at the true win rate (or at # sessions * true win rate), by Central Limit Theorem.

Btw, OP's interpretation of confidence intervals is way off.

Winrate is not a random variable because of the skill factor. But for someone playing constant stakes against constant opponent types using constant strategy and style, a bunch of equal-sized samples should follow a near-normal distribution. But that's a theoretical ideal, not real life.

How does "skill factor" imply winrate can't be modeled as a random variable? Do you leave each session at exactly 4.1 bb/100?

Are all the trials independent or do you adjust to your opponents over time? Do you improve over time? Do your opponents adjust to you? Do you adjust to your bankroll? Do they? And so forth.

That's why I emphasized model, I'm aware of what you wrote. Plus, one could just consider their last 100 sessions.

This is a tough one for me to figure out....lets see.....how do I find out if I am a winning poker player?......welllllll, I check my bank account every month....if it goes up, I am a winner.....if it goes down....I lost.

seriously though, I know math plays a large part of game, but I prefer to use it to make better decisions while playing.....then I will check my bank account! LOL

The hubris of the mathematically inept never ceases to amuse me.

Well, if someone were to be playing in a house with their friends, would that not be real life? If you understand different poker player types, then you know how to play against opponents. If you have the same players at your table, and you know how to play against them..but they do not know and do not care to learn how they should be playing against you..then you have more constants than variables, do you not?

Perhaps it would be better to calculate each hand as a sample instead of each 100 hands, then you could more accurately know what your standard deviation and standard error were because a lot of data goes missing when you add and subtract money won and lost in each hand? This would show the differences in the accuracy of how much a tight player wins/loses versus how much a loose player wins/loses. That way you would get 100 samples per session and could then know your average win/loss per hand played. That average win/loss per hand played could have a standard deviation and error, and then once found could be converted to a BB/100 win rate.

Why do you think that my interpretation of confidence intervals is way off? The confidence interval is set up to display with what confidence, based on the mean of a sample population, the true mean is. If I have the mean BB/100 per session, the standard deviation from the mean each session's BB/100 was, and the standard error of the sample size. Why can I not use that to set up a confidence interval with an upper and lower bound that indicates there is (using 1.96 as the critical value) 95% confidence that the true win rate is between my lower and upper bound? Is that not the purpose of confidence intervals?..

Saying "w/ 95% probability etc" is the wrong interpretation of confidence intervals. A 95% confidence interval means that if you repeat this experiment, 95% of confidence intervals will contain the true mean. But you can't say w/ 95% probability your interval will capture the true mean, a constant, it's either in there (probability 100%) or it's not (0%).

If this is indeed a Confidence Interval, and 95% of CI's will contain the true mean, then isn't there a 95% chance that this is one of the CI's that contains the true mean?

I asked my professor once, "If you create 100 95% confidence intervals and select one at random, then there is a 95% chance of getting a confidence interval w/ the true mean, right?" She hesitantly said yes.

But after I passed the class, I realized this is very iffy. If I create, say, 203 CI's, then still there's 95% chance of getting CI w/ the true mean when I select one at random. But then, if I create 1 CI and select one at "random," then saying there's a 95% chance of getting the true mean is a little weird and almost seems a contradiction for the reason stated in the last post. I suppose the statement still holds if you don't know the bounds of the CI.

Do you want me to email some professors?

No - the bolded part is exactly what you can say. Before you calculate an X% CI, you can say the probability that the interval will include or cover the true but unknown mean is X%. What you theoretically cannot say, is that after you have an X% confidence interval that the probability the true mean lies within the interval is X%.

But, the question then is -- just what does the confidence interval tell you about the true mean? We say the CI is an estimate. What is it an estimate of? Clearly it is an estimate of the mean, but it's interpretation is somewhat ambiguous. One way to think of it as follows: If, as OP said, the 95% CI for his win rate is (-3, 35) then he can be 95% 'confident' that the mean is somewhere in that interval. How's that for an enlightening statement?

There is another way to look at it for those who are uncomfortable with ascribing some type of degree of belief (probability, likelihood, confidence, etc.) to a confidence interval. If someone offered you a bet that OP was not a winning player, would his confidence interval result have any bearing on you taking the bet? To be more specific, would you be more likely to take the bet if the interval was (-3, 35), as it actually was, then if it was (-10, 10)?

OK...I went to a special school back in the 6th-7th grade for being mathematically gifted, and I have no damn idea what is going on here. Granted I am now 27, and I smoked a LOT of weed in between then and now... anyways, I just now got back into school and I am taking a math class which will be the same **** I learned when I was 12 years old, at best, but I seriously don't remember any of it. I gave up on school a LONG time ago. My question is this. Can someone please direct me to where I can learn 1. equity 2. EV 3. ICM, and whatever other kinds of math are used during a given session? I know odds and probability and all that basic **** like the back of my hand, but I want to start learning the advanced strategies. I also realize I can just go Google all of this, but that is so random and iffy.

On another note, why can't you simply keep a record of time played each session, your +/- for that session, no. of hands per hour, and so forth. Is this not in depth enough? I know if I looked back and saw that I was ahead $14,500 over 500 hours and had played x number of hands, I could come up with $ won per hr. and per session, etc. I would think (personally, no hating) that all the time spent on this would be better spent studying my game and improving my leaks...no? Again, this is all just questions, me trying to get an idea of where you guys are coming from. Thanks.

What you didn't put in bold is:

"A 95% confidence interval means that if you repeat this experiment, 95% of confidence intervals will contain the true mean"

So is that ^ wrong? And why?

How is that contradicting "Before you calculate an X% CI, you can say the probability that the interval will include or cover the true but unknown mean is X%"?